The Valley of the Dark Lords
Objectives: Gain prestige to become this year's Sith Apprentice
There are exits from this area to the Shyrack Caves, the Tomb of Ajunta Pall, the Tomb of Marko Ragnus, the Tomb of Tulak Hord, and the Tomb of Naga Sadow.
You'll run into Dak Vesser near the entrance to the Valley proper; if Juhani is in your party, he and she will have a frigid little back-and-forth which causes Dak to run off after you're done speaking to him. He'll still fill you in on the Tombs, if you ask nicely.
Lashowe is at the far north end of the tombs. When you meet up with her, a fight against a few tuk'ata will commence; kill them and Lashowe will obtain the holocron. She will, to no one's great surprise, attempt to run off to Uthar and gain the credit for herself. If you insist on presenting the holocron to him together, she'll attack you; kill her and take the holocron for yourself. Don't worry--this is a neutral Force alignment result.
Other than that, the four tombs here present good methods for Light side Jedi to obtain prestige without Dark side alignment shifts.
Objectives: Investigate the renegade Sith
The renegade Sith that you learned of earlier are sequestered away in the Shyrack caves, which is the first exit from the path that leads into the Valley of the Dark Lords. Once you track them down, you can, as always, kill them or help them. Helping them entails killing a terentatek in the eastern part of the caves. Whether you help them or destroy them, however, you'll want to fight through the terentatek anyway, since you'll find the bodies of the two other Jedi sent to kill the beasts to the north of its position, along with the incredible Qel-Droma Robes, which only a Light side Jedi can use.
Whether you kill the students, or help them escape, you'll be able to return to Uthar for prestige upon completion of this quest. If you let them live, lie to him by stating that the students are "gone."
The Tomb of Ajunta
Objectives: Find the Sword of Ajunta
The Tomb of Ajunta Pall is located on the eastern side of the Valley; head in and start looking around, if you can get past the locked door that begins the area (DC 28). Once inside, check the first Sith body you find for a demolitions sensor, which grants you +8 to Awareness and +4 to Demolitions; should be handy for the many mines you will hopefully not be stumbling across.
After you're inside, start picking through the bodies for datapads and other items. There'll be a pillar blocking your way that brings up an inventory screen when you click on it; put an explosive device, like a frag mine, inside of it to blow it up. The droids across the way will all suddenly activate and start firing upon you; destroy droid will shut them down with a minimum of fuss, as will fire from an ion gun. You can run to the far end of the path and pull the lever to kill them easily, but you won't gain any experience from doing so.
After another locked door (DC 28), you'll be inside Ajunta's Tomb. Click on the tomb door, and then the sarcophagus to gain three swords, and release the spirit of Ajunta himself. Talk to him, then place the notched steel sword onto the Sith Statue behind the tomb. Speak to Ajunta again; you'll get his true sword, and you'll also be able to attempt to turn his spirit back to the Light side for a large alignment adjustment. If you choose the wrong sword, or anger Ajunta during your dialogues, he'll attack you.
You'll run into Shaardan on your way out of the tomb, eager to "relieve you of your burden." You can give him one of the false swords, or force him into a fight and kill him. Or, heck, you can even give him the real sword if you wish; let it not be said that KOTOR doesn't give you options. If you do give him a false sword, he'll be killed by Uthar the next time you enter the Academy.